Sample Question 9:

A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16$

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Answer Keys

Question 9: D

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Solutions

Question 9
Step 1: Note that if the point $(2,3)$ is in the set $S$, then by symmetry along the line $y=x$, $(3,2)$ is also in $S$.

Step 2: By symmetry about the origin, the points $(-2,-3)$ and $(-3,-2)$ are also in $S$.

Step 3: By symmetry about the x-axis, the points $(2,-3)$ and $(3,-2)$ are in $S$.

Step 4: By symmetry about the y-axis, the points $(-2,3)$ and $(-3,2)$ are in $S$.

Step 5: Thus, every point of the form $(\pm 2, \pm 3)$ or $(\pm 3, \pm 2)$ must be in $S$, and all these 8 points satisfy all of the symmetry conditions. Therefore, the smallest number of points in $S$ is 8 (Option D).